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16
BASIC CHARACTERS OF THE UNITRIANGULAR GROUP (FOR ARBITRARY PRIMES)
, 2002
"... Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends ..."
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Cited by 18 (4 self)
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Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption p ≥ n, wherepisthe characteristic of the field Fq.
Nonsolvable Groups with No Prime Dividing Three Character Degrees
, 2010
"... Throughout this note, G will be a finite group, Irr(G) will be the set of irreducible characters of G, and cd(G) will be the set of character degrees of G. We consider groups where no prime divides at least three degrees in cd(G). Benjamin studied this question for solvable groups in [1]. She proved ..."
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Cited by 2 (2 self)
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Throughout this note, G will be a finite group, Irr(G) will be the set of irreducible characters of G, and cd(G) will be the set of character degrees of G. We consider groups where no prime divides at least three degrees in cd(G). Benjamin studied this question for solvable groups in [1]. She proved that solvable groups with this property satisfy cd(G)  � 6. She also presented examples to show
Character degree graphs that are complete graphs
 Proc. AMS 135
, 2007
"... Abstract. Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) −{1}, and there is an edge between a and b if (a, b)> 1. We prove that if Γ(G) is a complete graph, then G is a solvable group. 1. ..."
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Cited by 2 (2 self)
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Abstract. Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) −{1}, and there is an edge between a and b if (a, b)> 1. We prove that if Γ(G) is a complete graph, then G is a solvable group. 1.
HECKE ALGEBRAS FOR THE BASIC CHARACTERS OF THE UNITRIANGULAR GROUP
, 2003
"... Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a previous paper we obtained a decomposition of the regular character of Un(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ..."
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Cited by 1 (1 self)
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Let Un(q) denote the unitriangular group of degree n over the finite field with q elements. In a previous paper we obtained a decomposition of the regular character of Un(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ξ�(ϕ) ofUn(q). We prove that ξ�(ϕ) is induced from a linear character of an algebra subgroup of Un(q), and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of ξ�(ϕ) as characters induced from an algebra subgroup of Un(q). Finally, we identify a special irreducible constituent of ξ�(ϕ), which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption p ≥ n where p is the characteristic of the field) that gives a necessary and sufficient condition for ξ�(ϕ) to have a unique irreducible constituent.
Frobenius polytopes
 PREPRINT, HTTP://WWW.REED.EDU/ DAVIDP/HOMEPAGE/MYPAPERS/FROB.PDF (2004) BAUMEISTER, HAASE, NILL, AND PAFFENHOLZ
, 2004
"... A real representation of a finite group naturally determines a polytope, generalizing the wellknown Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general Frobenius group. ..."
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Cited by 1 (0 self)
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A real representation of a finite group naturally determines a polytope, generalizing the wellknown Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general Frobenius group.
c ○ Copyright by Maria I. Loukaki, 2001NORMAL SUBGROUPS OF ODD ORDER MONOMIAL P A Q BGROUPS BY
, 2004
"... A finite group G is called monomial if every irreducible character of G is induced from a linear character of some subgroup of G. One of the main questions regarding monomial groups is whether or not a normal subgroup N of a monomial group G is itself monomial. In the case that G is a group of even ..."
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A finite group G is called monomial if every irreducible character of G is induced from a linear character of some subgroup of G. One of the main questions regarding monomial groups is whether or not a normal subgroup N of a monomial group G is itself monomial. In the case that G is a group of even order, it has been proved (Dade, van der Waall) that N need not be monomial. Here we show that, if G is a monomial group of order p a q b, where p and q are distinct odd primes, then any normal subgroup N of G is also monomial. iii to my parents Anna and Giannis, and to my teacher, Everett C. Dade, without whom none of this would have been done, and everything would have been written faster. iv Acknowledgments First and foremost I would like to thank my advisor Prof. Everett C. Dade not only for his endless patience, his continued support and encouragement, his creativity and his humor, but also for all those afternoon meetings in his office, where I saw how mathematics can become pure art. Thank
IRREDUCIBLE CHARACTERS OF FINITE ALGEBRA GROUPS
, 1998
"... Let p be a prime number, let q = p e (e ≥ 1) be a power of p and let Fq denote the finite field with q elements. Let A be a finite dimensional Fqalgebra. (Throughout the paper, all algebras are supposed to have an identity element). Let J = J(A) be the Jacobson radical of A and let ..."
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Let p be a prime number, let q = p e (e ≥ 1) be a power of p and let Fq denote the finite field with q elements. Let A be a finite dimensional Fqalgebra. (Throughout the paper, all algebras are supposed to have an identity element). Let J = J(A) be the Jacobson radical of A and let
IRREDUCIBLY REPRESENTED GROUPS
, 2006
"... Abstract. A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infin ..."
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Abstract. A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented. We indicate some consequences of this for operator algebras. In particular, we charaterise up to isomorphism the countable subgroups ∆ of the unitary group of a separable infinite dimensional Hilbert space H of which the bicommutants ∆ ′ ′ (in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on H. 1. Gaschütz Theorem for infinite groups, and consequences Define a group to be irreducibly represented if it has a faithful irreducible unitary representation and irreducibly underrepresented 1 if not. For example, a finite abelian group is irreducibly represented if and only if it is cyclic (because finite subgroups of multiplicative groups of fields, in particular finite subgroups of C ∗ , are cyclic). It is a straightforward consequence of Schur’s lemma that a group of which the centre contains a non–cyclic finite subgroup is irreducibly underrepresented. For finite groups, there are also standard